Neutrino signals in electron-capture storage-ring experiments

Neutrino signals in electron-capture decays of hydrogen-like parent ions P in storage-ring experiments at GSI are reconsidered, with special emphasis placed on the storage-ring quasi-circular motion of the daughter ions D in two-body decays P -->D + e-neutrino. It is argued that to the extent that daughter ions are detected, these detection rates might exhibit modulations with periods of order seconds, similar to those reported in the GSI storage-ring experiments for two-body decay rates [1,2]. New dedicated experiments in storage rings, or using traps, could explore these modulations.


Introduction
Electron capture (EC) rates for two-body decays P → D + ν e of H-like parent ions P coasting in the GSI storage ring exhibit time modulation with laboratory period τ GSI ≈ 7 s [1,2] as shown for 142 Pm in Fig. 1. Here, the number of EC decays per time interval dt is given, in obvious notation, by with modulation angular frequency ω EC , corresponding to extremely minute energy splitting ω EC ≈ 0.83 × 10 −15 eV in the parent rest frame. The amplitude A of this modulation was found in the first experiment to be A=0.23±0.04 [1]. However, half as small amplitudes were obtained recently with improved sensitivity and time resolution of order 0.5 s [2]. The decay of parent ions in the GSI experiments is signaled, to a variable degree, by the correlated observation of daughter ions which are also confined to a storage-ring motion. In contrast to these storage-ring results, non storagering experiments elsewhere found no trace of EC decay-rate modulations [3,4]. A new storage-ring experiment has been proposed recently at GSI and is under active consideration for running by the end of 2014 [5]. Several works, beginning with the first GSI report [1], argued that the observed modulation might arise from interference between EC amplitudes A ν j that encode quantum entanglement of the daughter ion with the two mass-eigenstate neutrinos ν j (j = 1, 2) dominantly coupled to the electronneutrino ν e which is emitted in the EC two-body decay. Indeed, the neutrino on-shell energies E ν j (p j ) in the P → D + ν j decay rest frame differ by where ∆m 2 ν ≡ m 2 2 − m 2 1 = (0.750 ± 0.020) · 10 −4 eV 2 is the squared-mass difference of these two mass-eigenstate neutrinos [6] and M P ≈ 132 GeV is the rest mass of 142 Pm. The closeness of the energy splitting E ν 2 (p 2 )−E ν 1 (p 1 ) to ω EC inspired Ivanov and Kienle (IK) to attempt a rigorous derivation of ∆m 2 ν /2M P as the underlying energy splitting in two-body EC decays. Their paper [7] drew criticism from many theorists for the following two points: • No modulation is possible without interference. The IK derivation implements interference between two mass-eigenstate neutrino amplitudes A ν j (P → D + ν j ; t) which are summed up coherently, |A ν 1 + A ν 2 | 2 . Ivanov and Kienle [7] justified adding up amplitudes coherently by invoking momentum and energy measurement uncertainties to make up for the difference in momentum and in energy between the values assumed by each of the mass-eigenstate neutrinos ν j . But since masseigenstate neutrinos are distinct particles and, furthermore, are not observed in EC measurements, shouldn't these amplitudes get summed up incoherently, |A ν 1 | 2 + |A ν 2 | 2 , rather than coherently? This point was made particularly strongly in Refs. [8,9], see also the recent discussion by Peshkin [10].
• Careful inspection of the IK derivation leads to ω osc ≡ ∆m 2 ν /2E ν rather than ω IK ≡ ∆m 2 ν /2M P of Eq. (2) as the relevant energy splitting scale [11]. This is in accord with traditional discussions of neutrino oscillations [12]. Since E ν is of order several MeV in two-body EC decays, ω osc ≫ ω EC by roughly four orders of magnitude. Detecting modulation with angular frequency ω osc requires time resolution shorter by several orders of magnitude than available at present in the GSI storage-ring experiments [1,2]. Therefore, interference of the ECdecay mass-eigenstate neutrino amplitudes A ν j , even if allowed, cannot explain the reported modulation of EC decay rates.
The present note is focused on studying the time evolution of daughter ions D in the storage-ring spatially confined quasi circular motion. It is motivated by observing that the GSI technique of recording parent decay events is biased by the time evolution of daughter ions over a time interval of order 0.5 s during which they undergo rapid electron cooling. This aspect of the GSI experiment has been overlooked in previous studies and is considered here for the first time. It is worth noting that no compelling argument has ever been given against coherence between daughter-ion evolution amplitudes disentangled from their respective ν j neutrino propagation amplitudes. Such coherence could be sustained to some extent by the daughter ions upon coasting continuously in the storage ring, at least until completing the cooling cycle that enables their proper registration in the GSI measurements. It is shown here that the time evolution of daughter ions in storage-ring motion gives rise to oscillations with angular frequency that under certain conditions is similar to that given by Eq. (2), close to the modulation frequency reported by the GSI experiments. This may be tested by keeping track of the steady motion of daughter ions in the storage ring under stable electron cooling for as long as possible. In this way, the weight with which daughter ions affect the decay-rate measurement could be varied and thereby evaluated.
The paper is organized as follows. Neutrino-mass scenarios for deriving modulation angular frequencies of EC decay rates are briefly revisited in Sect. 2, assuming that coherence can be justified. We confront the IK result ω IK = ∆m 2 ν /2M P [7] with the more natural one ω osc = ∆m 2 ν /2E ν [11], both evaluated in the parent-ion rest frame, and discuss the reason for this difference. Oscillations arising from the time evolution of daughter ions are discussed in Sect. 3, giving rise now to angular frequency ω ∼ ∆m 2 ν /2 M P in the same frame, close to the reported ω EC [1,2]. This new result is specific to storage-ring quasi-circular confined motion of ions and does not hold in rectilinear two-body decays. Special care has to be exercised in transforming this angular frequency to the laboratory frame and apply it to the experimental conditions prevailing in the GSI storage-ring experiments, as discussed in Subsect. 3.1. Finally, daughter-ion oscillations are derived in Subsect. 3.2 in complete analogy to traditional formulations of neutrino oscillations [12]. The paper ends with a brief Conclusion section.

Modulation of two-body EC decay rates revisited
In the IK construction [7], the amplitude A ν j (P → D+ν j ; t) for recording at time t EC decays that occurred at any time τ from injection time τ = 0 to observation time τ = t is expressed by an with daughter-ion D and ν j -neutrino energies E D and E ν j , respectively, and momentum q in the parent-ion P rest frame. Integrating on τ leads to and a decay rate given for sufficiently long time t by Summing incoherently the ν j -neutrino induced rates R ν j gives with unitary mixing matrix U defined by |ν α = 3 j=1 U * αj |ν j , α = e, µ, τ . By integrating over the final cm momentum q, the magnitude q j of the onshell momentum that satisfies the δ(∆ j ( q j )) constraint in the jth term is picked up. This dependence on j can be safely ignored, as done here. The incoherent sum of terms (6) must be supplemented by j ′ = j interference terms, amounting to A crucial observation to make in evaluating (7) is that it is impossible to satisfy both δ(∆ j ) and δ(∆ j ′ ) simultaneously with the same cm momentum vector q, so the phase (∆ j −∆ j ′ )t is to be evaluated once under the constraint ∆ j ( q j ) = 0 and once under the constraint ∆ j ′ ( q j ′ ) = 0. On each occasion: where both δ(∆ j ) and δ(∆ j ′ ) are evaluated at the same momentum, the precise value of which in fact makes no difference as long as it is within the momentum measurement uncertainties. The period of the resulting modulation factor in (7) is hence T = 2πω osc −1 ∼ 4 · 10 −4 s for E ν ∼ 4 MeV. Given a time resolution of order 0.5 s in the GSI experiment, this modulation averages to zero in the EC decays observed at GSI.
The constraints imposed by δ(∆ j ) + δ(∆ j ′ ) in (7) on the evaluation of the interference phase (∆ j −∆ j ′ )t were ignored by IK who used on-shell momenta IK argued that their choice of on-shell p j and p j ′ that differ by a tiny amount, is legitimate because the difference (10) is considerably smaller than the uncertainties incurred in momentum measurements. Numerically, however, one finds that the IK result ∆m 2 jj ′ /2M P (9) is unstable against replacing p j and p j ′ , independently of each other, by distributed values of wavepacket momenta, leading thereby to a modulation angular frequency closer to ∆m 2 jj ′ /2E ν as given by (8). For example, interchanging p j and p j ′ on the l.h.s. of (9) yields We conclude that even if it were allowed to sum up coherently neutrino-mass eigenstate amplitudes A ν j (P → D + ν j ; t) in the evaluation of two-body EC decay rates, the modulation arising from interfering amplitudes would correspond to angular frequencies of order ω osc which disagree by roughly four orders of magnitude with the GSI reported angular frequency ω EC [1,2]. Finally, we note that no special features specific to circular storage-ring motion, as opposed to rectilinear motion of parent ions, were deemed necessary to apply in this derivation (following Ref. [11]) of parent-ion decay rates in the parent-ion rest frame P. To transform from time t in P to time t L in the laboratory frame L one uses as usual the Lorentz factor γ: t L = γt.

Daughter-ion oscillations in two-body EC
The calculation of the two-body EC decay rate outlined in the previous section does not address the time evolution of the daughter-ion D and the electron-neutrino ν e which are produced entangled with each other. As stated in the Introduction, the GSI technique of extracting EC decay rates is biased by the time evolution of the daughter ion over time intervals of order 0.5 s. This aspect of the GSI experiment has been overlooked in past calculations and is considered here for the first time.

Effects of storage-ring confined motion
A similarly entangled two-body system consists of a positively charged muon and muon-neutrino ν µ produced in the two-body weak decay of a positively charged pion, π + → µ + + ν µ . It was shown recently by Kayser [12] that for a rectilinear motion of such a two-body system (say, in the pion rest frame) and up to order O(∆m 2 ν ), the time evolution of the µ + probability density exhibits no modulation or oscillation superimposed on the normal weak-interaction exponential decay rate exp(−Γ µ t). This should be contrasted with the time evolution of the emergent neutrino probability density which exhibits oscillations with angular frequency commensurate with ω osc of Eq. (8). Such oscillations, say ν µ → ν e , are observable in principle by detecting an electron-neutrino ν e at a sufficiently distant terrestrial point. The GSI experiment differs in one essential respect from any gedanken experiment looking for µ + -recoil oscillations, as well as from EC experiments done with rectilinear geometry [3,4]. In the GSI experiment the daughter ion follows the same storage-ring quasi-circular trajectory traversed by the decaying parent ion. We now outline how the evaluation of daughter-ion storage-ring motion differs from that of µ + -recoil rectilinear motion in Ref. [12].
In this approach the space-time motion of the stable daughter ion D is described ideally by a plane wave per each entangled neutrino species j. For times of order seconds, as in the GSI experiment, the neutrino wave packet is peaked at x values far away from those spanned by D. This allows us to disregard the neutrino undetected trajectory when considering the storage-ring motion of D, except implicitly through the dependence of the plane wave (12) on the neutrino species j. In the π + → µ + +ν µ rectilinear motion analysis, the phase factor ( p µj · x−E µj t), analogous to the phase factor for Dj in (12), was shown to be independent of j up to order O(∆m 2 ν ), so that up to this order the time evolution of the outgoing µ + incurs no modulation or oscillation [12]. This is achieved by requiring that the space-time coordinates x and t follow the motion of the muon wave packet, whereby the j dependence of p µj · x exactly cancels the j dependence of E µj t. Although demonstarted in the pion (parent) rest frame, owing to Lorentz covariance of the space-time phase in (12) this holds in any Lorentz-related frame, including the laboratory frame.
In the present case, because D is confined to the storage ring, it can be shown that effects arising from the phase p Dj · x are negligible. Although each phase p Dj · x is not necessarily small the difference of two such phases, recalling the magnitude of p j − p j ′ from (10) and using a spatial size of x ∼ 50 m, is negligible, |(p Dj − p Dj ′ )x|≪1, and will be disregarded henceforth in considering potential contributions to interference terms. Suppressing therefore p Dj · x spatial phases, a nonvanishing j dependence emerges from the phase E Dj t. Assuming that no further j dependence arises from the gone-away neutrino, and using (2), the interference term between ψ Dj (x ≈ 0, t) and ψ Dj ′ (x ≈ 0, t) for j = j ′ is given in the parent rest frame P by 2 cos(∆E (P ) where ∆E (P ) and the rest-frame time t P = 0 corresponds to the EC decay time. Note the absence here of the δ-function constraints from (7) that would have ruled out the emergence of the on-shell energies E (P ) Dj and E (P ) Dj ′ . This latter distinction is of crucial importance, as the oscillation angular frequency reached in (13) is identical with ∆m 2 ν /2 M P of Eq. (2), coming close to the GSI value of ω EC . However, this is not the end of the story since we need to evaluate these interference contributions in the lab frame L.
In the lab frame L, the plane wave (12) assumes the form which upon suppressing the spatial phase p (L) Dj · x L gives rise to a lab interference term 2 cos(∆E (L) with oscillation angular frequency that corresponds to a lab energy splitting ∆E Dj ′ . This is evaluated by transforming from the rest frame P to the lab frame L, where ∆ p (P ) Dj ′ and β = 0.71 with γ ≡ (1 − β 2 ) −1/2 = 1.42 [1,2]. This Lorentz transformation holds everywhere on the storage-ring trajectory of D. Recalling from (2) and (10) that |∆E it could be argued that the first term on the r.h.s. of (16) may be safely neglected with respect to the second one which by (10) is of order ∆m 2 ν /2E ν , several orders of magnitude larger than ω EC . This renders unobservable the oscillations of daughter ions coasting in the GSI storage ring when electron cooling, which reduces primarily the longitudinal momenta of D, is switched off. The effect of electron cooling, upheld at present for approximately 0.5 s from EC decay, is that the daughter-ion circular motion follows a revolution frequency identical to that obtained by imposing the transversality condition β · p (P ) Dj = 0. This suggests, but does not necessarily imply, that β ·∆ p (P ) Djj ′ = 0 too holds to the required accuracy. If it does, then ∆E (L) As long as electron cooling of the daughter ions did not destroy the coherence input of ∆E Djj ′ , the resulting cooled D storage-ring motion should exhibit this coherence, expressed now by Eq. (17). For completeness we note that ∆p Djj ′ too comes out as small, ∆p (L) Djj ′ = βγ∆m 2 j ′ j /2M P , using a similar reasoning.
Substituting ∆E

(L)
Djj ′ from (17) in the interference term (15) leads to the following period of daughter-ion oscillations in the lab frame L: where the uncertainty is due to the uncertainty in the value of ∆(m 2 ν ). Compared with the reported value of 7.12 ± 0.11 s [2], we miss just a factor (10.17 ± 0.26)/(7.12 ± 0.11) = 1.43 ± 0.06 consistent with γ = 1.42. However, this coincidence might well be fortuitous. Minute departures from β · p (P ) Dj = 0 could make this term modify appreciably expression (17) for ∆E To demonstrate the sensitivity of the result (17) to reasonable departures from a strict β · p (P ) Dj = 0 assumption, we note that electron cooling affects mostly the large longitudinal momentum components p (L) (parallel to beam directionβ) and less so the small transverse components p (L) ⊥ = p (P ) sin θ P through small-angle Coulomb scattering. Retaining ∆p where in the last step we averaged over sin 2 θ P . This ∆E Djj ′ agrees within a factor of approximately 3 with that of (17). The main message is that the relevant lab energy difference ∆E

(L)
Djj ′ is of the order of ∆m 2 ν /2M, not of order ∆m 2 ν /2E ν . A comment is in order: one might have naively expected that a transition from P to L can be made directly in the argument of the cos functions in (13) simply upon substituting t P by t L /γ, thereby obtaining ∆m 2 jj ′ /2γ M P for the lab oscillation angular frequency. This procedure is invalid because the motion of frame P is confined and hence its space-time coordinates (x P , t P ) are not related correctly by any fixed Lorentz transformation to (x L , t L ) as long as the spatial motion is confined. This difficulty does not arise in calculating decay rates which require only the well tested t L = γt P relationship between the lab time t L and the proper time t P . Finally we note that daughter-ion oscillations start at time t = t d when parent-ions decay, whereas the modulations reported by the GSI experiments [1,2] are conceived to have started at time t = 0 defined by the injection of parent ions into the storage ring. This, as well as a different evaluation of daughter-ion oscillations which connects directly to the entangled neutrino, is discussed in the following subsection.

Analogy to neutrino oscillations
As argued in the previous subsection, daughter-ion oscillations may be evaluated in the two-body decay P → D + ν e in full analogy with the way neutrino oscillations are evaluated in π + → µ + + ν µ . Following standard derivations, e.g. [12], the amplitude A D for the two-body EC production of D at time t d and its detection at time t is given in the parent rest frame P by where τ = t − t d and j denotes the daughter-ion component Dj entangled with the neutrino mass eigenstate ν j . The apearance of the same Dj component in both production and detection follows from the orthogonality of the entangled neutrino mass eigenstates ν j . The restriction to the electron flavor e in the detection vertex follows from the small phase space of a few MeV available to the emitted neutrino if/once it too was detected. Note that the mass-eigenstate neutrinos have been projected out within a standard wave-packet treatment, thereby disentangling the Dj components from the neutrinos and allowing for oscillations. To be specific we specialize to a 2 × 2 orthogonal neutrino mixing matrix with angle θ, expressing the probability |A D | 2 as which exhibits the same oscillation/modulation angular frequency ∆E discussed in the previous subsection. This expression does not yet give the differential rate for EC decay marked by detection of the daughter ion. One needs to integrate (21) over τ P between 0 to t P , allowing thereby the decay time to span a temporal range from injection to observation, and then differentiate by t P to get the required rate. Since τ P and t P are linearly related, the final result for the rate is practically the same as given by (21): with ∆E (P ) D12 = ∆m 2 ν /2M P , see Eq. (13) and the text that follows it. Expression (22) exhibits modulations with amplitude relative to a base line of 1 − sin 2 (2θ)/2 ≈ 0.572 ± 0.012, where the value sin 2 (2θ) = 0.857 ± 0.024 [6] was used. This summarizes our prediction for future experiments focusing on daughter-ion oscillations in confined motion.

Conclusion
In this note we have reconsidered the modulation of EC decay rates reported by GSI storage-ring studies [1,2] in terms of neutrino signals. Repeating arguments published in [11] it was shown in Sect. 2 that if and once coherence between different mass-eigenstate neutrino EC decay amplitudes A ν j (P → D + ν j ; t) can be justified, EC decay rates should exhibit modulation with angular frequencies ω osc ∼ 10 4 s −1 , where ω osc = ∆m 2 ν /2E ν . This derivation does not use any special properties of storage-ring geometry and would hold also in more conventional EC-decay experimental studies, e.g. [3,4]. Such modulation cannot be observed in the GSI experiments at present.
In the second part of this note, Sect. 3, we derived a nonvanishing oscillation frequency that a daughter ion confined to the GSI storage ring should exhibit as long as remaining entangled with the neutrino produced in the two-body EC decay of a parent ion coasting in the storage ring. This frequency also invloves the neutrinos squared-mass difference ∆m 2 ν , now divided by 2M P , commensurate with the GSI EC decay rate modulation frequency ω EC ∼ 1 s −1 . It is specific to storage-ring motion, vanishing in rectilinear daughter-ion motion as discussed in Ref. [12]. The precise value of the period T (L) osc given under idealized conditions by Eq. (18) depends sensitively on the extent to which the neutrino and the daughter ion keep entangled to each other under given experimental constraints. The detection of daughter ions disentangles them from the neutrinos, allowing for daughter-ion oscillations.
As long as GSI storage-ring EC decay experiments do not separate between observables pertaining to decay parent ions and observables associated with the produced daughter ions, daughter-ion oscillations will persist in giving rise to apparent modulation of EC decay rates, with amplitudes that depend on how the decay of parent ions P and the subsequent storage-ring motion of daughter ions D are both taken into account in the experimental analysis. This subject obviously requires more work which is outside the scope of this note.